A graph $G$ is said to be \emph{determined by its spectrum} if any graph having the same spectrum as $G$ is isomorphic to $G$. Let $K_n \setminus P_{\ell}$ be the graph obtained from $K_n$ by removing edges of $P_\ell$, where $P_\ell$ is a path of length $\ell-1$ which is a subgraph of a complete graph $K_n$. Cámara and Haemers~\cite{MC} conjectured that $K_n \backslash P_{\ell}$ is determined by its adjacency spectrum for every $2\leq \ell \leq n$. In this paper we show that the conjecture is true for $7\leq \ell \leq9$.